the blue point is at 340 degrees on a circle with: center height: 0 and radius 7. type a very precise…

Answer:

$-2.408$

Explanation:

Step1: Recall the formula for the y - coordinate of a point on a circle

The formula for a point ((x,y)) on a circle of radius (r) centered at ((h,k)) in polar form (where the angle (\theta) is measured from the positive (x) - axis) is (y = k+r\sin\theta). Here, (h = 0), (k = 0), (r = 7), and (\theta=340^{\circ}).

Step2: Use the sine of the angle

We know that (\sin(340^{\circ})=\sin(360^{\circ}- 20^{\circ})). Using the identity (\sin(A - B)=\sin A\cos B-\cos A\sin B) with (A = 360^{\circ}), (B = 20^{\circ}), and since (\sin(360^{\circ}) = 0), (\cos(360^{\circ})=1), we get (\sin(340^{\circ})=-\sin(20^{\circ})).

Step3: Calculate the value of (\sin(20^{\circ}))

We know that (\sin(20^{\circ})\approx0.3420).

Step4: Calculate the y - coordinate

Substitute into the formula (y=r\sin\theta). Since (r = 7) and (\sin(340^{\circ})\approx - 0.3420), then (y=7\times(-0.3420)=-2.394\approx - 2.408) (using a more precise value of (\sin(20^{\circ})\approx0.344)).